4.4 Further Topics in Regression Analysis

Objectives:By the end of this section, I will be able to…

1) Explain prediction error, calculate SSE, and utilize the standard error s as a measure of a typical prediction error.

2) Describe how total variability, prediction error, and improvement are measured by SST, SSE, and SSR.

3) Explain the meaning of r2 as a measure of the usefulness of the regression.

Regression Analysis

Analysts use correlation and linear regression to analyze a data set.

They also look at the data and determine “errors”.

PREDICTION ERROR

Measures how far the predicted value, is from the actual value, y, observed in the data set.

Sum of Squares Error (SSE)

All the prediction errors are squared and then added up.

Standard Error of the Estimate, s

A measure of the size of the typical prediction error.

Total Sum of Squares, SST

A measure of the total variability in the values of the y variable.

Sum of Squares Regression, SSR

Measures the amount of improvement in the accuracy of our estimates when using the regression equation compared with relying only on the y values and ignoring the x information.

Combined Relationship

SST = SSR + SSE

Coefficient of Determination, r2

Measures the goodness of fit of the regression equation to the data.

It is the ratio of SSR/SST. Is between 0 and 1.

Data Set

Volume, x Weights, y

4 10

8 16

12 25

16 30

20 35

Find the following values

1. Regression Line

2. r

3. SSE

4. s

5. SSR

6. SST

7. r2

Data Set Volume

xWeights

yPredicted

scoreResidual Residual2

4 10

8 16

12 25

16 30

20 35

Data Set Volume

xWeights

yPredicted

scoreResidual Residual2

4 10

8 16

12 25

16 30

20 35

To find the predicted score we have to find the regression line using our calculators.

y = 4 + 1.6x

10.4

16.8

23.2

29.6

36

10-10.4-0.4

16-16.8 -0.8

(-0.4)20.16

(-0.8)2

(1.8)2

(0.4)2

(-1)2

0.64

3.24

0.16

1

10-23.2-13.2

16-23.2

25-23.2

-7.2

1.8

(-13.2)2

(-7.2)2

(1.8)2

(6.8)2

(11.8)2

174.24

51.84

3.24

46.24

139.24

(10.4-23.2)2

(16.8-23.2)2

(23.2-23.2)2

(29.6-23.2)2

(36-23.2)2

163.84

40.96

0

40.96

163.84

SSRSSTSSE5.2 414.8 409.6

25-23.2

30-29.6 30-23.2

35-23.235-36

1.8

6.8

11.8

0.4

-1

Regression Line, r, and r2 were found on calculator.

There is only one left to find…

1.316561177

Time to get a Program! Then find the following

values for the data set.

1. Regression Line

2. r

3. SSE

4. s

5. SSR

6. SST

7. r2

HEIGHTS (in inches)

HAND LENGTH (in inches)